Download SP17 Lecture Notes 6 - Confidence Interval for a Population Mean

Lecture notes 6 – confidence
interval for a population mean
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Confidence interval outline
The point estimate
The margin of error
Interpretation of a confidence interval
Worked example
Confidence Intervals
We have discussed the idea of using a point estimate
(which is a statistic calculated from observed data) to
estimate the value of a parameter (which is unknown).
For instance, we may use a sample mean (a point
estimate) to estimate the value of a population mean (a parameter).
One issue we face is that our estimates are nearly
always wrong. And they are wrong not because we
make mistakes (though this is possible), but because
they are based on limited, random samples. A sample
mean, for instance, is random insofar as the sample
itself is random. If a new sample is taken, a different
mean will be calculated.
Confidence Intervals
So how do we deal with the fact that we are nearly
always wrong? By trying to determine how wrong we
may be. Perhaps we believe that our estimate is
probably only off by a little. Perhaps we believe that
our estimate is likely to be off by quite a bit.
We quantify this by calculating a margin of error,
which we add and subtract from our point estimate in
order to form a confidence interval (CI):
CI = point estimate ± margin of error
The point estimate
The point estimate is our “best guess” for the value of
an unknown parameter. It is a sample statistic that we
calculate from our data.
For instance,
is a point estimate for
.
It is always important to keep in mind that, while a
point estimate is our best guess, there will always been
some “error” in this guess. It is for this reason that we
find margins of error to go along with point estimates.
The margin of error
The margin of error is the maximum amount by which
we believe our point estimate may be in error (i.e. by
which is may differ from the true unknown parameter
value), at a desired “level of confidence” or “confidence
level”.
The margin of error will be found by multiplying a
critical value by a standard error. We then add and
subtract this from the point estimate. Visually, this
process is akin to drawing a t-distribution centered on
the point estimate, and then chopping off the tails:
Interpretation of a CI
When we build a confidence interval, we expect that it
will capture the unknown parameter value. You can
think of the CI as a range of plausible values for the
unknown parameter, at some level of confidence.
The level of confidence is the % chance that a CI
constructed using a random sample will capture the
true unknown parameter value. So, if we make a 95%
CI for a parameter, then we know that, 95% of the
time, these kinds of CIs will capture the parameter.
However we never get to know if it really did capture it
this time.
Example
In many animal species, individuals communicate
through UV signals visible to one another but invisible
to humans. A scientist is interested in the role of UV
colors in the Lissotriton vulgaris newt, and conducted a
study that measured the difference in length of time a
female of the species spent near males with and without
the UV presence. A positive measurement indicates
that the female spent longer time with the UV present,
and a negative means less time under the same
conditions.
The average measurement from 23 trials is 50.7, with a
standard deviation of 87.3. Construct a 99% CI for the
true average time difference given by all newts in this
scenario.
Formula for a confidence interval
All of the confidence intervals we will see follow this
general formula:
CI for unknown parameter =
point estimate ± margin of error
And the margin of error (ME) will follow the general
formula:
ME = (critical value) * (standard error of point estimate)
So, we have the general formula:
CI = point estimate ± (critical value) * (standard error)
Formula for a confidence interval
To apply this general formula to the case of constructing
a confidence interval for a population mean, note that
the point estimate is the sample mean, the critical value
is in terms of the t-distribution, and the standard error
of the sample mean is the sample standard deviation
divided by the square root of the sample size. And so
we have:
Note that “t” is a critical value, which is found on your ttable under the column for a 99% CI.
Constructing the CI
Using this formula, we can construct the CI in the
space below:
Interpreting the CI
Once the CI is constructed, we need to give an English
interpretation. The general form of some acceptable
interpretations is:
“The range of plausible values for ___________, at ____%
confidence, is _______ to _______.”
“We are _____% confident that the true ____________ lies
between ________ and ________.”
“Under repeated sampling, _____% of similarly
constructed intervals will contain the true __________.”
Interpreting the CI
Note that we do not say “There is a ____% chance that the true ________ lies
between _______ and _______.”
This is because the value of the parameter is fixed (but
unknown). Once we have constructed a CI, the
parameter either is or is not contained in this interval.
We just don’t get to know whether it is or is not.
The key here is that the CI is the thing that is random,
not the parameter. We can talk about the probability
that a random CI will contain the parameter, not the
probability that a specific CI contains the parameter; it
either does or does not.
Interpreting the CI
Give an interpretation of the CI we constructed in the
space below: